# In-Context Algebra

arXiv: https://arxiv.org/abs/2512.16902 · HTML: https://mlsanity.com/paper/2512.16902/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 71 of its 88 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 30 of 71)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task](https://mlsanity.com/paper/2210.13382/) (2022)
- [In-Context Learning Creates Task Vectors](https://mlsanity.com/paper/2310.15916/) (2023)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [Direct and Indirect Effects](https://mlsanity.com/paper/1301.2300/) (2013)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations](https://mlsanity.com/paper/2303.02536/) (2023)
- [Causal Abstraction: A Theoretical Foundation for Mechanistic Interpretability](https://mlsanity.com/paper/2301.04709/) (2023)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Fine-Tuning Enhances Existing Mechanisms: A Case Study on Entity Tracking](https://mlsanity.com/paper/2402.14811/) (2024)
- [Pretraining task diversity and the emergence of non-Bayesian in-context learning for regression](https://mlsanity.com/paper/2306.15063/) (2023)
- [Analyzing Transformers in Embedding Space](https://mlsanity.com/paper/2209.02535/) (2022)
- [Sudden Drops in the Loss: Syntax Acquisition, Phase Transitions, and Simplicity Bias in MLMs](https://mlsanity.com/paper/2309.07311/) (2023)
- [How do Language Models Bind Entities in Context?](https://mlsanity.com/paper/2310.17191/) (2023)
- [Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process](https://mlsanity.com/paper/2407.20311/) (2024)

## Built on by (most cited first, 5 of 5)

- [Interpreting and Steering State-Space Models via Activation Subspace Bottlenecks](https://mlsanity.com/paper/2602.22719/) (2026)
- [When Symbol Names Should Not Matter: A Logistic Theory of Fresh-Symbol Classification](https://mlsanity.com/paper/2605.07120/) (2026)
- [Data-Efficient Adaptation of LLMs via Attention Head Reweighting](https://mlsanity.com/paper/2607.13425/) (2026)
- [Causal and Interpretable Structures in LLM Compositional Tasks](https://mlsanity.com/paper/2609.35970/) (2026)
- [Conditional Transfer from Controlled Pretraining Mixtures to Code](https://mlsanity.com/paper/2610.11548/) (2026)

## Abstract (arXiv)

We investigate the mechanisms that arise when transformers are trained to solve arithmetic on sequences where tokens are variables whose meaning is determined only through their interactions in-context. While prior work has studied transformers in settings where the answer relies on fixed parametric or geometric information encoded in token embeddings, we devise a new in-context reasoning task where the assignment of tokens to specific algebraic elements varies from one sequence to another. Despite this challenging setup, transformers achieve near-perfect accuracy on the task and even generalize to unseen groups. We develop targeted data distributions to create causal tests of a set of hypothesized mechanisms, and we isolate three mechanisms models consistently learn: commutative copying where a dedicated head copies answers, identity element recognition that distinguishes identity-containing facts, and closure-based cancellation that tracks group membership to constrain valid answers. Our findings show that the kinds of reasoning strategies learned by transformers are dependent on the task structure and that models can develop symbolic reasoning mechanisms when trained to reason in-context about variables whose meanings are not fixed.